Probability of Error With a Parity Error$-$Detecting Code

Eight data bits and one parity bit form one codeword of an error$-$detecting code: if$,$ at the receiver, the parity bit does not match the data bits, a request for re$-$transmission of this codeword is sent to the source.

Assume each bit is received correctly with probability $0\mathrm{.}99,$ independently of other bits. That is to say, there is a $1\%$ chance that any bit may be incorrectly received $$ including the parity bit.

Define events True Positive, False Positive, True Negative, and False Negative as shown below. Note that a False Positive event occurs when only the parity bit is received incorrectly:

some or all data bits incorrectly received all $8$ data bits correctly received

parity error detected

True Positive False Positive

no parity error detected

False Negative True Negative

Answer these questions and show all of your work:

$1.$ What is the probability that the parity bit is received incorrectly?

$2.$ What is the probability that all $8$ data bits are received correctly?

$3.$ What is the probability that exactly one of the $8$ data bits is received incorrectly?

$4.$ What is the probability that $\{1,3,5,$ or $7\}$ data bits are received incorrectly?

$5.$ What is the probability that $\{2,4,6,$ or $8\}$ data bits are received incorrectly?

$6.$ What is the probability of event "True Positive"?

$7.$ What is the probability of event "True Negative"?

$8.$ What is the probability of event "False Positive"?

$9.$ What is the probability of event "False Negative"?

$10.$ What is the probability that the parity test will give a Negative $($no parity error$)$ result?

$11.$ What is the probability that the received parity bit correctly indicates data integrity?

Include the case where the parity bit is received incorrectly.